66 THEORY OF COLLINEATIONS. 
is called a projective transformation or collineation of the 
plane. Using homogeneous point coordinates the same trans- 
formation is expressed by the linear equations, 
px, = ax-+ by + cz, py,=a'x+b'y+c'z, pz,=a"a-+ by + ce’. 
(2) 
We shall generally use the Cartesian system, but may occa- 
sionally use the other form. The change from the one sys- 
tem to the other is so easily made that the reader will have 
no difficulty in passing from the one to the other at pleasure. 
We shall assume throughout, unless otherwise expressly 
stated, that the coefficients and variables in equations (1) 
and (2) are complex numbers; we shall also assume that the 
determinant of the transformation does not vanish; thus 
Boe 
al! 1b el! 
The reason for excluding, for the present, transformations 
for which the determinant vanishes will be shown later when 
these special transformations with determinant equal to zero 
will be discussed. 
77. One-to-one Correspondence of Points. Equations (1) 
show that «, and y, are one-valued functions of x and y, or 
that to a given point (,y) there corresponds in a collineation 
one and only one point (#,,y,). Equations (1) can be solved 
for « and y; the values found are 
Me Ag+ A'’yit+ A” 
Cai+Cyi+ C” 
where A, A’, B, ete., are the cofactors of a, a’, b, ete., in A, 
the determinant of the transformation (1). 
The solvability of equations (1) is secured by assuming that 
A is not zero. Equations (3) are likewise one-valued func- 
tions of x, and y,. Hence to any chosen point (2,,y,) there 
corresponds one and only one point (,y). 
Equations (1) transform the point (2, y) into (w,,y,) ; while 
equations (3) transform the point (#,,y,) back to (a,y). Two 
on = #0. 
Ba+ B’yit+ B” 
Cm 4 Cy Cl (3) 
and Yy a Cai+Cyi+ C” , 
